Archive for posts tagged ‘sports’
In Business
Kevin Day, August 15th, 2008As of July 2nd, I am no longer employed by General Electric. Also, I am also no longer a mechanical engineer.
What have I been doing with myself?
I have been voraciously developing the best fantasy football software website on the internet.
Fantasy Football Calculator is now hosting 1,500 live mock drafts every day. I’ve also launched three new products within the past two months: the Lineup Calculator, Draft Calculator, and Draft Simulator.
To keep up with the high traffic volume during the NFL pre-season, I’ve had to scale the site from one server to four. That process wasn’t fun, but the site is now running smoothly even during peak hours.
I’ve also started a blog on that site, which is another reason I’ve been neglecting this one. I’m going to start posting here again because I do have non-fantasy football related thoughts now and then.
Sweet sixteen probabilities
Kevin Day, March 23rd, 2008Thanks to my fiancée, Kristen, for proposing this interesting question: What is the probability that someone has a perfect bracket just through the first two rounds this year?
First, as a baseline, the odds of there being no upsets in the first two rounds of the Men’s NCAA Basketball Tournament are one in 13.2 million. This was calculated with the same winning model as used in my last post.
The odds of the first two rounds occuring like it has so far in the 2008 tournament are one in 376 billion.
Like I showed the other day, that doesn’t mean that the odds are 1 in 376 billion that someone would have correctly predicted the first two rounds. Without doing all of the simulations, I would guess that the odds would increase by about 10,000 because that’s how much they improved last time from the random model to the smarter model.
That means the probability of a perfect bracket through the two rounds so far this year are about one in 37.6 million.
[Edited to give Kristen proper credit for the question]
The Perfect NCAA Bracket
Kevin Day, March 22nd, 2008In grade school, my favorite game was “Starting Lineup Talking Baseball”, which simulated baseball games based on the players you had on your team. The success of each player depended on their real-life statistics, so it was important to memorize everyone’s stats.
The direct correlation of the players’ statistics to their success in that game may be why I think sports are deterministic and why I love trying to predict things about sports.
For a few years now, I’ve been thinking about the NCAA tournament. There are a few simple questions that I don’t think have been appropriately answered. This post answers the question, “What are the odds of picking the perfect bracket?”
Spoiler
Here’s the answer if you’re in a hurry and don’t care about the details. The chance that your bracket is perfect is:
1 in 924 trillion
… so you’re saying there’s a chance?
Well, put it this way. If 10 million people fill out brackets this year, there’s only a 1 in 100 million chance that anyone will have a perfect bracket.
Traditional Answer
The common way that I’ve seen this question answered (WSJ, Math Forum) is that the total number of possible brackets are calculated as 2^63, (neglecting the play-in game) which is 9.22 * 10^18. Then they go on to say that the chance of getting a perfect bracket is one in 9.22 * 10^18.
However, this assumes that every bracket is equally probable. Usually studies mention this caveat, but they don’t really take the next step and find out what the real probabilities are.
The Wall Street Journal article linked to above makes some simplifications about how accurate people guess winners and how many games can be considered a lock for the favorite, but in my opinion they’re still just guesses.
A better way to find the probability of the bracket being perfect is to first model the probability of each seed winning a game. Then, use that model to calculate the probabilities of different selection strategies people might use.
Modeling with Tournament Seeds
There have been a few studies that have modeled the relationship between a team’s tournament seed and their probability of winning a game (Carlin; Schwertman, et al). One of the simplest and most accurate models is
P(Team A) = Team B’s Seed / (Team A’s Seed + Team B’s Seed)
This says that if a #2 seed plays a #6 seed, the chance that the #2 seed will win is 6 / (2 + 6), which is 75%. The chance the #6 seed will win is 2 / (2 + 6), which is 25%.
This model does a good job representing historical win probabilities in the NCAA tournament, except for when #1 seeds play #2 seeds. Historically, #1 seeds only win 43% of the time instead of 67% predicted by the model (Schwertman, et al).
If you adjust for the #1 vs. #2 error, the model can be used to estimate the probability that any bracket will be perfect. All you have to do is go game by game through the bracket multiplying the win probabilities of each predicted winner. The result is the chance that every predicted winner in the entire bracket will actually win.
Below I calculate the probabilities of perfect brackets for five different strategies for selecting winners.
All Favorites
This strategy picks all higher-ranked seeds. At the Final Four, each team has an equal shot of winning since they’re all #1 seeds, so there’s actually 8 possible All Favorites brackets.
1 in 546,000,000,000
Most Favorites
This strategy is the one used by nearly everyone when they pick their bracket. Most of the higher seeds win, but there are several upsets as well.
I’m not making any distinction between strategies such as “teams hot at the end of the year are better” or “teams with big guys inside do better in the tournament.” Any bracket that choses favorites to win as frequently as they have in past NCAA tournaments fall within this strategy.
Since there can be many different Most Favorites brackets, I generated 8,000 brackets and took the average probability of all of them.
1 in 924,000,000,000,000
Random
This strategy assumes each team has an equal chance of winning. Again, I generated 8,000 brackets and took the average probability.
1 in 308,000,000,000,000,000,000
Most Upsets
This strategy is the opposite of the of the Most Favorites strategy. Underdogs are favored in most of the games, but higher-ranked teams win occasionally. Number 16 seeds are frequently chosen to win the championship with this strategy. This is the average probability of 8,000 brackets.
1 in 893,000,000,000,000,000,000,000,000,000
All Upsets
The opposite of the All Favorites strategy is picking 100% upsets. That’s right, your Final Four is all #16 seeds. What are the odds of that happening?
1 in 144,000,000,000,000,000,000,000,000,000,000
Conclusion
Not all brackets are created equal. A bracket with all upsets is 10 sextillion times less likely to occur than one with all favorites.
Unless you honestly have a legitimate way of predicting winners better than most people, the odds of your bracket being perfect are 1 in 924 trillion. Although not very good, these odds are 10,000 times better than the number usually reported. All right!
However, your odds could be a thousand times better if you just picked all of the higher seeds to win. Why don’t you?
First of all, it’s not fun. Most people like to take risks and then feel proud when they correctly predict upsets.
But I think there are other reasons as well. For instance, if everyone in your league picked all favorites, then someone would have the exact same bracket as you and you would have a lower chance of winning the league.
Or would you?
My next post will look into the trade-offs between these strategies when competing in a league. There’s some game-theory type questions that I’m not qualified to address but will try to answer anyways.
Stay tuned.
